A250033
a(n) gives the denominators for A250031(n) as well as for A250032(n).
Original entry on oeis.org
1, 2, 2, 24, 15, 45, 35, 280, 630, 2100, 2310, 27720, 10010, 140140, 150150, 480480, 255255, 2297295, 4849845, 96996900, 101846745, 106696590, 111546435, 669278610, 929553625, 966735770, 1003917915, 3123300180, 3234846615, 48522699225, 100280245065, 1604483921040, 6618496174290, 6819056664420, 7019617154550, 7220177644680
Offset: 1
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s_aux(n,p0,inp)={my(t=0/1,tt=0/1,in=inp,pp);while(1,pp=p0*prime(in);tt=n\pp;if(tt==0,break,t+=tt/pp-s_aux(n,pp,in++)));return(t)};
s(n)=1+s_aux(n,1,1);
a=vector(1000,n,denominator(s(n-1)/n))
A072155
Denominator of Sum_{k=1..n} phi(k)/k.
Original entry on oeis.org
1, 2, 6, 3, 15, 5, 35, 70, 210, 210, 2310, 770, 10010, 10010, 30030, 15015, 255255, 255255, 4849845, 4849845, 4849845, 4849845, 111546435, 37182145, 37182145, 37182145, 111546435, 111546435, 3234846615, 3234846615, 100280245065, 200560490130, 200560490130
Offset: 1
1, 3/2, 13/6, 8/3, 52/15, 19/5, 163/35, 361/70, 1223/210, ...
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List([1..35], n-> DenominatorRat( Sum([1..n], k-> Phi(k)/k) ) ); # G. C. Greubel, Aug 25 2019
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[Denominator( &+[EulerPhi(k)/k: k in [1..n]] ): n in [1..35]]; // G. C. Greubel, Aug 25 2019
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with(numtheory); seq(denom( add(phi(k)/k, k=1..n)), n =1..35); # G. C. Greubel, Aug 25 2019
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Table[Sum[EulerPhi[k]/k,{k,n}],{n,40}]//Denominator (* Harvey P. Dale, Jun 08 2017 *)
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a(n) = denominator(sum(k=1, n, eulerphi(k)/k)); \\ Michel Marcus, Jan 26 2015
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[denominator( sum(euler_phi(k)/k for k in (1..n)) ) for n in (1..35)] # G. C. Greubel, Aug 25 2019
A250031
a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))=1.
Original entry on oeis.org
0, 1, 1, 13, 8, 26, 19, 163, 361, 1223, 1307, 16477, 5749, 83977, 88267, 280817, 147916, 1377406, 2839897, 58552633, 60492571, 63263911, 65468386, 403117367, 549883871, 579629587, 596790577, 1864736021, 1912541636, 29293503812, 59449633388, 969992016739
Offset: 1
When n=10, the density of numbers m that are coprime to floor(m/10) turns out to be 1223/2100. Hence a(10) = 1223/2100.
When n=2, all odd numbers qualify, but only the m=2 among even numbers does; hence the density is 1/2 and therefore a(2)=1.
When n=1, only m=1 qualifies, so that the density is 0, and a(1) = 0.
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s_aux(n,p0,inp)={my(t=0/1,tt=0/1,in=inp,pp);while(1,pp=p0*prime(in);tt=n\pp;if(tt==0,break,t+=tt/pp-s_aux(n,pp,in++)));return(t)};
s(n)=1+s_aux(n,1,1);
a=vector(1000,n,numerator(1-s(n-1)/n))
A250034
Numerators a(n) of the rational-valued function s(n) defined below.
Original entry on oeis.org
1, 3, 11, 7, 38, 16, 117, 269, 877, 1003, 11243, 4261, 56163, 61883, 199663, 107339, 1839778, 2009948, 38444267, 41354174, 43432679, 46078049, 1064644972, 379669754, 387106183, 407127338, 1258564159, 1322304979, 38458390826, 40830611677, 1268983808602
Offset: 1
n=4: s(4) = 1 - (-1)*(floor(4/2)/2 + floor(4/3)/3) = 1 + 1 + 1/3 = 7/3, with a(4) = 7 and 3 is indeed A072155(4). - _Wolfdieter Lang_, Dec 02 2014
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s_aux(n,p0,inp)={my(t=0/1,tt=0/1,in=inp,pp);while(1,pp=p0*prime(in);tt=n\pp;if(tt==0,break,t+=tt/pp-s_aux(n,pp,in++)));return(t)};
s(n)=1+s_aux(n,1,1);
a=vector(1000,n,numerator(s(n)))
Showing 1-4 of 4 results.
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